The internal circuitry that supports power and data to and from the externally enclosed
hard disk drive may likely have a failure in one or another connection or hardware part.
So to remove the hard drive from the enclosure and attach it more directly in attempt
to retrieve the content is a suggested method to try & recover information, since there
is a fair chance the hard drive itself is not the main problem. It may be OK enough to
use a universal USB cable kit to at least test the theory, since it isn't terribly expensive.
Sometimes the format used in an old external drive may not be suitable for Mac OS X
but may be OK for some kinds of files or to be shared with a Windows PC; however
the sound of your issue appears to point to a failure to communicate between device
and computer. A bad cable or printed circuit inside the enclosure, a partially failed
power supply inside the enclosure, & so on. And the old hard drive may be worn out.
Hopefully you can at least recover the content of the old hard drive; be sure that any
computer you use to recover to, has a known-good additional external HDD attached
so the data won't have to be saved into the computer directly. Especially if the files
may over-fill the computer's internal hard drive.
Good luck in this matter! 🙂